5.
Process Improvement
5.5.
Advanced topics
5.5.4.
What is a mixture design?
5.5.4.2.
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Simplex-lattice designs
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| Definition
of simplex-lattice points |
A {q, m} simplex lattice
design for q components consists of points defined by the following
coordinate settings: the proportions assumed by each component take the
m+1
equally spaced values from 0 to 1, |
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xi = 0, 1/m,
2/m, …, 1 for i = 1, 2, …, q |
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and all possible combinations
(mixtures) of the proportions from this equation are used. |
| Except for
the center, all design points are on the simplex boundaries |
Note that the standard Simplex-Lattice
and the Simplex-Centroid designs (described later) are boundary point designs;
that is, with the exception of the overall centroid, all the design points
are on the boundaries of the simplex. When one is interested in prediction
in the interior, it is highly desirable to augment the simplex type designs
with interior design points. |
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Consider a three component
mixture where the number of equally spaced levels for each component is
four (i.e. xi = 0, 0.333, 0.667, 1). In this example then q
= 3 and m = 3. If one considers all possible blends of the three
components with these proportions, then the {3, 3} simplex-lattice contains
the 10 blending coordinates listed in the table below. The experimental
region and the distribution of design runs over the simplex region is shown
in the figure below. There are a total of 10 design runs for the {3, 3}
simplex lattice design. |
| Example
of a 3-component simplex lattice design |
TABLE 5.3 Simplex Lattice Design
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FIGURE 5.9 Configuration of Design Runs for a {3,3}
Simplex-Lattice Design
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The number of design points
in the simplex-lattice is (q+m-1)!/m!(q-1)!. |
| Definition
of canonical polynomial model used in mixture experiments |
Now consider the form of the
polynomial model that one might fit to the data from a mixture experiment.
Due to the restriction x1 + x2 + ... + xq
= 1, the form of the regression function that is fit to the data from mixture
experiments is somewhat different than the traditional polynomial fit and
is often referred to as the canonical polynomial. Its form is derived using
the general form of the regression function that can be fit to data collected
at the points of a {q, m} simplex-lattice design and substituting
into this the dependence relationship among the xi terms. The
number of terms in the {q, m} polynomial is (q+m-1)!/m!(q-1)!.
This number is equal to the number of points that make up the associated
{q, m} simplex-lattice design. |
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For example, the equation that
can be fit to the points from a {q, m=1} simplex-lattice design
is |
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 |
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This is called the canonical
form of the first order mixture model. In general, the canonical forms
of the mixture models (with the asterisks removed from the parameters)
are as follows: |
| Summary
of canonical mixture models |
 |
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The terms in the canonical
mixture polynomials have simple interpretations. Geometrically, the parameter
bi
in the above equations represent the expected response to the pure mixture
xi=1, xj=0, ij, and is the height of the mixture
surface at the vertex xi=1. The portion of each of the above
polynomials given by |
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is called the linear blending
portion. When blending is strictly additive, then the linear model form
above is an appropriate model. |
| Three component
mixture example |
The following example is from
Cornell (1990) and consists of a three component mixture problem. The three
components are Polyethylene (X1), polystyrene (X2), and polypropylene (X3)
which are blended together to form fiber that will be spun into yarn. The
product developers are only interested in the pure and binary blends of
these three materials. The response variable of interest is yarn elongation
in kilograms of force applied. A {3,2} simplex-lattice design is used to
study the blending process. The simplex region and the six design runs
are shown in the figure below. The figure was generated in JMP version
3.2. The design and the observed responses are listed in the table below.
There were two replicate observations run at each of the pure blends. There
were three replicate observations run at the binary blends. There are a
total of 15 observations with six unique design runs. |
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FIGURE 5.10 Design Runs for the {3,2} Simplex-Lattice Yarn
Elongation Problem
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TABLE 5.4 Simplex-Lattice Design for
Yarn Elongation Problem
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The design runs listed in the
above table are in standard order. The actual order of the 15 treatment
runs was completely randomized. JMP 3.2 will be used to analyze the results.
Since there are three levels of each of the three mixture components, a
quadratic mixture model can be fit to the data. The output from the model
fit is shown below. Note that there was no intercept fit in the model.
To analyze the data in JMP, create a new table with one column corresponding
to the observed elongation values. Select Fit Model and create the quadratic
mixture model (this will look like the 'traditional' interactions regression
model obtained from standard classical designs). Check the No Intercept
box on the Fit Model screen. Click on Run Model. The output is shown below. |
| JMP analysis
for the mixture model example |
JMP Output for {3,2} Simplex-Lattice Design
Screening Fit
Summary of Fit
RSquare
0.951356
RSquare Adj
0.924331
Root Mean Square Error
0.85375
Mean of Response
13.54
Observations (or Sum Wgts) 15
Analysis of Variance
Source DF
Sum of Squares Mean Square
F Ratio
Model
5 128.29600
25.6592
35.2032
Error
9
6.56000 0.7289
C Total 14
134.85600
Prob > F < .0001
Tested against reduced model: Y=mean
Parameter Estimates
Term Estimate
Std Error t Ratio
Prob>|t|
X1
11.7
0.603692 19.38
<.0001
X2
9.4 0.603692
15.57 <.0001
X3
16.4
0.603692 27.17
<.0001
X2*X1 19
2.608249 7.28
<.0001
X3*X1 11.4
2.608249 4.37
0.0018
X3*X2 -9.6
2.608249 -3.68
0.0051
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Under the parameter estimates
section of the output are the individual t-tests for each of the parameters
in the model. The three cross product terms are significant (X1*X2, X3*X1,
X3*X2), indicating a significant quadratic fit. |
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The fitted quadratic mixture
model is |
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 |
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Since b3 > b1
>
b2, one can conclude that component 3 (polypropylene) produces
yarn with the highest elongation. Additionally, since b12 and b13 are positive,
blending components 1 and 2 or components 1 and 3 produce higher elongation
values than would be expected just by averaging the elongations of the
pure blends. This is an example of 'synergistic' blending effects. Components
2 and 3 have antagonistic blending effects because b23 is negative. |
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The figure below is the contour
plot of the elongation values From the plot it can be seen that if maximum
elongation is desired, a blend of components 1 and 3 should be chosen consisting
of about 75% - 80% component 3 and 20% - 25% component 1. |
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FIGURE 5.11 Contour Plot of Predicted Elongation Values from
{3,2} Simplex-Lattice Design
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